Step 1
Remove cores from tomatoes and cut in half crosswise. Reserve until needed.
Step 2
Heat olive oil and butter in a saucepan over medium-high heat. Add garlic and anchovy and cook until garlic is sizzling in the hot fat 45 seconds to 1 minute. Carefully add the tomatoes with their juices salt and red pepper flakes. Toss with a spoon every few minutes until the tomatoes release their liquid and start to break down.
Step 3
Once the tomatoes have softened and start to collapse reduce heat to medium-low and cook stirring occasionally until the tomatoes have fallen apart and the soup resembles a coarse tomato sauce about 45 minutes.
Step 4
Meanwhile preheat the oven to 450 degrees F (230 degrees C). Line a baking sheet with foil or a silicone liner. Drizzle with 2 tablespoons olive oil.
Step 5
Place the bread slices on the prepared baking sheet. Top each slice with sliced basil and then Cheddar cheese.
Step 6
Bake in the preheated oven until the cheese is well browned and crispy around the edges 15 to 20 minutes. Let cool to room temperature.
Step 7
Remove soup from heat and pass through a mesh strainer into a bowl. Use the back of a ladle to push all the tomato mixture through. Discard the skins and seeds which should be the only things left in the strainer.
Step 8
Transfer the strained tomato soup back into the saucepan and place back over medium-low heat. Continue cooking to thicken slightly if desired or simply bring back to a simmer and serve immediately with the cheese toasts.